Precalculus Examples

(1 + cot (x)) (1 - cot (x)) - {csc}^{2} (x)

$(1 + \cot (x)) (1 - \cot (x)) - \csc^{2} (x)$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Expand

(1 + cot (x)) (1 - cot (x))

$(1 + \cot (x)) (1 - \cot (x))$ using the FOIL Method.

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Step 1.1.1.1

Apply the distributive property.

1 (1 - cot (x)) + cot (x) (1 - cot (x)) - {csc}^{2} (x)

$1 (1 - \cot (x)) + \cot (x) (1 - \cot (x)) - \csc^{2} (x)$

Step 1.1.1.2

Apply the distributive property.

1 \cdot 1 + 1 (- cot (x)) + cot (x) (1 - cot (x)) - {csc}^{2} (x)

$1 \cdot 1 + 1 (- \cot (x)) + \cot (x) (1 - \cot (x)) - \csc^{2} (x)$

Step 1.1.1.3

Apply the distributive property.

1 \cdot 1 + 1 (- cot (x)) + cot (x) \cdot 1 + cot (x) (- cot (x)) - {csc}^{2} (x)

$1 \cdot 1 + 1 (- \cot (x)) + \cot (x) \cdot 1 + \cot (x) (- \cot (x)) - \csc^{2} (x)$

1 \cdot 1 + 1 (- cot (x)) + cot (x) \cdot 1 + cot (x) (- cot (x)) - {csc}^{2} (x)

$1 \cdot 1 + 1 (- \cot (x)) + \cot (x) \cdot 1 + \cot (x) (- \cot (x)) - \csc^{2} (x)$

Step 1.1.2

Simplify and combine like terms.

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Step 1.1.2.1

Simplify each term.

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Step 1.1.2.1.1

Multiply

1

$1$ by

1

$1$ .

1 + 1 (- cot (x)) + cot (x) \cdot 1 + cot (x) (- cot (x)) - {csc}^{2} (x)

$1 + 1 (- \cot (x)) + \cot (x) \cdot 1 + \cot (x) (- \cot (x)) - \csc^{2} (x)$

Step 1.1.2.1.2

Multiply

- cot (x)

$- \cot (x)$ by

1

$1$ .

1 - cot (x) + cot (x) \cdot 1 + cot (x) (- cot (x)) - {csc}^{2} (x)

$1 - \cot (x) + \cot (x) \cdot 1 + \cot (x) (- \cot (x)) - \csc^{2} (x)$

Step 1.1.2.1.3

Multiply

cot (x)

$\cot (x)$ by

1

$1$ .

1 - cot (x) + cot (x) + cot (x) (- cot (x)) - {csc}^{2} (x)

$1 - \cot (x) + \cot (x) + \cot (x) (- \cot (x)) - \csc^{2} (x)$

Step 1.1.2.1.4

Rewrite using the commutative property of multiplication.

1 - cot (x) + cot (x) - cot (x) cot (x) - {csc}^{2} (x)

$1 - \cot (x) + \cot (x) - \cot (x) \cot (x) - \csc^{2} (x)$

Step 1.1.2.1.5

Multiply

- cot (x) cot (x)

$- \cot (x) \cot (x)$ .

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Step 1.1.2.1.5.1

Raise

cot (x)

$\cot (x)$ to the power of

1

$1$ .

1 - cot (x) + cot (x) - ({cot}^{1} (x) cot (x)) - {csc}^{2} (x)

$1 - \cot (x) + \cot (x) - (\cot^{1} (x) \cot (x)) - \csc^{2} (x)$

Step 1.1.2.1.5.2

Raise

cot (x)

$\cot (x)$ to the power of

1

$1$ .

1 - cot (x) + cot (x) - ({cot}^{1} (x) {cot}^{1} (x)) - {csc}^{2} (x)

$1 - \cot (x) + \cot (x) - (\cot^{1} (x) \cot^{1} (x)) - \csc^{2} (x)$

Step 1.1.2.1.5.3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

1 - cot (x) + cot (x) - {cot (x)}^{1 + 1} - {csc}^{2} (x)

$1 - \cot (x) + \cot (x) - {\cot (x)}^{1 + 1} - \csc^{2} (x)$

Step 1.1.2.1.5.4

Add

1

$1$ and

1

$1$ .

1 - cot (x) + cot (x) - {cot}^{2} (x) - {csc}^{2} (x)

$1 - \cot (x) + \cot (x) - \cot^{2} (x) - \csc^{2} (x)$

1 - cot (x) + cot (x) - {cot}^{2} (x) - {csc}^{2} (x)

$1 - \cot (x) + \cot (x) - \cot^{2} (x) - \csc^{2} (x)$

1 - cot (x) + cot (x) - {cot}^{2} (x) - {csc}^{2} (x)

$1 - \cot (x) + \cot (x) - \cot^{2} (x) - \csc^{2} (x)$

Step 1.1.2.2

Add

- cot (x)

$- \cot (x)$ and

$\cot (x)$ .

$1 + 0 - \cot^{2} (x) - \csc^{2} (x)$

Step 1.1.2.3

Add

$1$ and

$0$ .

$1 - \cot^{2} (x) - \csc^{2} (x)$

Step 1.2

Simplify with factoring out.

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Step 1.2.1

Move

$- \csc^{2} (x)$ .

$1 - \csc^{2} (x) - \cot^{2} (x)$

Step 1.2.2

Reorder

$1$ and

$- \csc^{2} (x)$ .

$- \csc^{2} (x) + 1 - \cot^{2} (x)$

Step 1.2.3

Factor

$- 1$ out of

$- \csc^{2} (x)$ .

$- (\csc^{2} (x)) + 1 - \cot^{2} (x)$

Step 1.2.4

Rewrite

$1$ as

$- 1 (- 1)$ .

$- \csc^{2} (x) - 1 \cdot - 1 - \cot^{2} (x)$

Step 1.2.5

Factor

$- 1$ out of

$- \csc^{2} (x) - 1 \cdot - 1$ .

$- (\csc^{2} (x) - 1) - \cot^{2} (x)$

Step 2

Apply pythagorean identity.

$- \cot^{2} (x) - \cot^{2} (x)$

Step 3

Subtract

$\cot^{2} (x)$ from

$- \cot^{2} (x)$ .

$- 2 \cot^{2} (x)$